Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 50.4% → 99.9%

Time: 9.2s

Alternatives: 12

Speedup: 3.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y \cdot 2\right)\\ \frac{\mathsf{fma}\left(y, 2, x\right)}{t_0} \cdot \frac{x - y \cdot 2}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* y 2.0))))
   (* (/ (fma y 2.0 x) t_0) (/ (- x (* y 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (y * 2.0));
	return (fma(y, 2.0, x) / t_0) * ((x - (y * 2.0)) / t_0);
}

Julia

function code(x, y)
	t_0 = hypot(x, Float64(y * 2.0))
	return Float64(Float64(fma(y, 2.0, x) / t_0) * Float64(Float64(x - Float64(y * 2.0)) / t_0))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(N[(y * 2.0 + x), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 48.8%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 48.8%

      \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. difference-of-squares 48.8%

      \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   3. *-commutative 48.8%

      \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. associate-*r* 48.8%

      \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. sqrt-prod 48.8%

      \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   6. sqrt-unprod 20.2%

      \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   7. add-sqr-sqrt 34.4%

      \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   8. metadata-eval 34.4%

      \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   9. *-commutative 34.4%

      \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   10. associate-*r* 34.4%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   11. sqrt-prod 34.4%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   12. sqrt-unprod 20.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   13. add-sqr-sqrt 48.8%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   14. metadata-eval 48.8%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

4. Applied egg-rr 48.8%

   \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 48.8%

      \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

   2. times-frac 50.4%

      \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

   3. +-commutative 50.4%

      \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   4. fma-def 50.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   5. add-sqr-sqrt 50.4%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. hypot-def 50.4%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   7. sqrt-prod 21.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   8. *-commutative 21.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   9. sqrt-prod 21.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   10. metadata-eval 21.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   11. associate-*r* 21.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   12. add-sqr-sqrt 50.4%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   13. *-commutative 50.4%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

7. Final simplification 99.9%

   \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
8. Add Preprocessing

Alternative 2: 47.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{hypot}\left(x, y \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 2000:\\ \;\;\;\;\frac{x - y \cdot 2}{t_1} \cdot \left(1 + \frac{y \cdot 2}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{t_1}^{2}}{x + y \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{t_1} \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (hypot x (* y 2.0))))
   (if (<= t_0 2000.0)
     (* (/ (- x (* y 2.0)) t_1) (+ 1.0 (/ (* y 2.0) x)))
     (if (<= t_0 5e+210)
       (/ (fma y 2.0 x) (/ (pow t_1 2.0) (+ x (* y -2.0))))
       (* (/ (fma y 2.0 x) t_1) (+ (* 0.5 (/ x y)) -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = hypot(x, (y * 2.0));
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = ((x - (y * 2.0)) / t_1) * (1.0 + ((y * 2.0) / x));
	} else if (t_0 <= 5e+210) {
		tmp = fma(y, 2.0, x) / (pow(t_1, 2.0) / (x + (y * -2.0)));
	} else {
		tmp = (fma(y, 2.0, x) / t_1) * ((0.5 * (x / y)) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = hypot(x, Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 2000.0)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) / t_1) * Float64(1.0 + Float64(Float64(y * 2.0) / x)));
	elseif (t_0 <= 5e+210)
		tmp = Float64(fma(y, 2.0, x) / Float64((t_1 ^ 2.0) / Float64(x + Float64(y * -2.0))));
	else
		tmp = Float64(Float64(fma(y, 2.0, x) / t_1) * Float64(Float64(0.5 * Float64(x / y)) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$0, 2000.0], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+210], N[(N[(y * 2.0 + x), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * 2.0 + x), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e3

   1. Initial program 57.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 57.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 57.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 25.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 50.6%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 50.6%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 50.6%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 25.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 57.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.3%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.3%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 48.0%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.0%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 48.0%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   if 2e3 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e210

   1. Initial program 81.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 81.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 81.6%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 81.6%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 26.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 35.2%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 35.2%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 35.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 26.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 81.6%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. associate-/l* 82.0%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x - y \cdot 2}}} \]

      2. div-inv 82.0%

         \[\leadsto \color{blue}{\left(x + y \cdot 2\right) \cdot \frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x - y \cdot 2}}} \]

      3. +-commutative 82.0%

         \[\leadsto \color{blue}{\left(y \cdot 2 + x\right)} \cdot \frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x - y \cdot 2}} \]

      4. fma-def 82.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, x\right)} \cdot \frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x - y \cdot 2}} \]

      5. *-commutative 82.0%

         \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}}{x - y \cdot 2}} \]

      6. associate-*r* 82.0%

         \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}}{x - y \cdot 2}} \]

      7. metadata-eval 82.0%

         \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}}{x - y \cdot 2}} \]

      8. swap-sqr 82.0%

         \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}{x - y \cdot 2}} \]

      9. pow1 82.0%

         \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{\color{blue}{{\left(x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)\right)}^{1}}}{x - y \cdot 2}} \]

      10. metadata-eval 82.0%

          \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{{\left(x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}}{x - y \cdot 2}} \]

      11. sqrt-pow2 82.0%

          \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{\color{blue}{{\left(\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}\right)}^{2}}}{x - y \cdot 2}} \]

      12. hypot-udef 82.0%

          \[\leadsto \mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}}^{2}}{x - y \cdot 2}} \]

   6. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, x\right) \cdot \frac{1}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x - y \cdot 2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 82.0%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right) \cdot 1}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x - y \cdot 2}}} \]

      2. *-rgt-identity 82.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x - y \cdot 2}} \]

      3. *-commutative 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x - \color{blue}{2 \cdot y}}} \]

      4. cancel-sign-sub-inv 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{\color{blue}{x + \left(-2\right) \cdot y}}} \]

      5. metadata-eval 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x + \color{blue}{-2} \cdot y}} \]

      6. *-commutative 82.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x + \color{blue}{y \cdot -2}}} \]

   8. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x + y \cdot -2}}} \]

   if 4.9999999999999998e210 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 15.2%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 15.2%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 7.5%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 7.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 7.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 7.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 7.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 15.2%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 17.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 17.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 17.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in x around 0 55.6%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2000:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + \frac{y \cdot 2}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{\frac{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}{x + y \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x - y \cdot 2\\ t_2 := \mathsf{hypot}\left(x, y \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 2000:\\ \;\;\;\;\frac{t_1}{t_2} \cdot \left(1 + \frac{y \cdot 2}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{t_1 \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{t_2} \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (- x (* y 2.0))) (t_2 (hypot x (* y 2.0))))
   (if (<= t_0 2000.0)
     (* (/ t_1 t_2) (+ 1.0 (/ (* y 2.0) x)))
     (if (<= t_0 5e+210)
       (/ (* t_1 (+ x (* y 2.0))) (+ t_0 (* x x)))
       (* (/ (fma y 2.0 x) t_2) (+ (* 0.5 (/ x y)) -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = x - (y * 2.0);
	double t_2 = hypot(x, (y * 2.0));
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = (t_1 / t_2) * (1.0 + ((y * 2.0) / x));
	} else if (t_0 <= 5e+210) {
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (fma(y, 2.0, x) / t_2) * ((0.5 * (x / y)) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(x - Float64(y * 2.0))
	t_2 = hypot(x, Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 2000.0)
		tmp = Float64(Float64(t_1 / t_2) * Float64(1.0 + Float64(Float64(y * 2.0) / x)));
	elseif (t_0 <= 5e+210)
		tmp = Float64(Float64(t_1 * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(fma(y, 2.0, x) / t_2) * Float64(Float64(0.5 * Float64(x / y)) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$0, 2000.0], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+210], N[(N[(t$95$1 * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * 2.0 + x), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e3

   1. Initial program 57.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 57.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 57.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 25.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 50.6%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 50.6%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 50.6%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 25.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 57.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.3%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.3%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 48.0%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.0%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 48.0%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   if 2e3 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e210

   1. Initial program 81.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 81.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 81.6%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 81.6%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 26.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 35.2%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 35.2%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 35.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 26.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 81.6%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.9999999999999998e210 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 15.2%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 15.2%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 7.5%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 7.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 7.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 7.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 7.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 15.2%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 17.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 17.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 17.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in x around 0 55.6%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2000:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + \frac{y \cdot 2}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x - y \cdot 2\\ \mathbf{if}\;t_0 \leq 2000:\\ \;\;\;\;1 + \frac{-6}{{\left(\frac{x}{y}\right)}^{2}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{t_1 \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (- x (* y 2.0))))
   (if (<= t_0 2000.0)
     (+ 1.0 (/ -6.0 (pow (/ x y) 2.0)))
     (if (<= t_0 5e+210)
       (/ (* t_1 (+ x (* y 2.0))) (+ t_0 (* x x)))
       (* (/ t_1 (hypot x (* y 2.0))) (+ 1.0 (* 0.5 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = x - (y * 2.0);
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = 1.0 + (-6.0 / pow((x / y), 2.0));
	} else if (t_0 <= 5e+210) {
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (t_1 / hypot(x, (y * 2.0))) * (1.0 + (0.5 * (x / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = x - (y * 2.0);
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = 1.0 + (-6.0 / Math.pow((x / y), 2.0));
	} else if (t_0 <= 5e+210) {
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (t_1 / Math.hypot(x, (y * 2.0))) * (1.0 + (0.5 * (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = x - (y * 2.0)
	tmp = 0
	if t_0 <= 2000.0:
		tmp = 1.0 + (-6.0 / math.pow((x / y), 2.0))
	elif t_0 <= 5e+210:
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = (t_1 / math.hypot(x, (y * 2.0))) * (1.0 + (0.5 * (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(x - Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 2000.0)
		tmp = Float64(1.0 + Float64(-6.0 / (Float64(x / y) ^ 2.0)));
	elseif (t_0 <= 5e+210)
		tmp = Float64(Float64(t_1 * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(t_1 / hypot(x, Float64(y * 2.0))) * Float64(1.0 + Float64(0.5 * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = x - (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 2000.0)
		tmp = 1.0 + (-6.0 / ((x / y) ^ 2.0));
	elseif (t_0 <= 5e+210)
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	else
		tmp = (t_1 / hypot(x, (y * 2.0))) * (1.0 + (0.5 * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2000.0], N[(1.0 + N[(-6.0 / N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+210], N[(N[(t$95$1 * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e3

   1. Initial program 57.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 57.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 57.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 25.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 50.6%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 50.6%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 50.6%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 25.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 57.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.3%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.3%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 48.0%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.0%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 48.0%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   10. Taylor expanded in y around 0 77.0%

       \[\leadsto \color{blue}{1 + -6 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 77.0%

          \[\leadsto 1 + \color{blue}{\frac{-6 \cdot {y}^{2}}{{x}^{2}}} \]

       2. associate-/l* 77.0%

          \[\leadsto 1 + \color{blue}{\frac{-6}{\frac{{x}^{2}}{{y}^{2}}}} \]

       3. unpow2 77.0%

          \[\leadsto 1 + \frac{-6}{\frac{\color{blue}{x \cdot x}}{{y}^{2}}} \]

       4. unpow2 77.0%

          \[\leadsto 1 + \frac{-6}{\frac{x \cdot x}{\color{blue}{y \cdot y}}} \]

       5. times-frac 83.0%

          \[\leadsto 1 + \frac{-6}{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} \]

       6. unpow2 83.0%

          \[\leadsto 1 + \frac{-6}{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}} \]

   12. Simplified 83.0%

       \[\leadsto \color{blue}{1 + \frac{-6}{{\left(\frac{x}{y}\right)}^{2}}} \]

   if 2e3 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e210

   1. Initial program 81.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 81.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 81.6%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 81.6%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 26.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 35.2%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 35.2%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 35.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 26.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 81.6%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.9999999999999998e210 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 15.2%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 15.2%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 7.5%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 7.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 7.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 7.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 7.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 15.2%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 17.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 17.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 17.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 55.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2000:\\ \;\;\;\;1 + \frac{-6}{{\left(\frac{x}{y}\right)}^{2}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x - y \cdot 2\\ t_2 := \frac{t_1}{\mathsf{hypot}\left(x, y \cdot 2\right)}\\ \mathbf{if}\;t_0 \leq 2000:\\ \;\;\;\;t_2 \cdot \left(1 + \frac{y \cdot 2}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{t_1 \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (- x (* y 2.0)))
        (t_2 (/ t_1 (hypot x (* y 2.0)))))
   (if (<= t_0 2000.0)
     (* t_2 (+ 1.0 (/ (* y 2.0) x)))
     (if (<= t_0 5e+210)
       (/ (* t_1 (+ x (* y 2.0))) (+ t_0 (* x x)))
       (* t_2 (+ 1.0 (* 0.5 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = x - (y * 2.0);
	double t_2 = t_1 / hypot(x, (y * 2.0));
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = t_2 * (1.0 + ((y * 2.0) / x));
	} else if (t_0 <= 5e+210) {
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = t_2 * (1.0 + (0.5 * (x / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = x - (y * 2.0);
	double t_2 = t_1 / Math.hypot(x, (y * 2.0));
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = t_2 * (1.0 + ((y * 2.0) / x));
	} else if (t_0 <= 5e+210) {
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = t_2 * (1.0 + (0.5 * (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = x - (y * 2.0)
	t_2 = t_1 / math.hypot(x, (y * 2.0))
	tmp = 0
	if t_0 <= 2000.0:
		tmp = t_2 * (1.0 + ((y * 2.0) / x))
	elif t_0 <= 5e+210:
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = t_2 * (1.0 + (0.5 * (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(x - Float64(y * 2.0))
	t_2 = Float64(t_1 / hypot(x, Float64(y * 2.0)))
	tmp = 0.0
	if (t_0 <= 2000.0)
		tmp = Float64(t_2 * Float64(1.0 + Float64(Float64(y * 2.0) / x)));
	elseif (t_0 <= 5e+210)
		tmp = Float64(Float64(t_1 * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(t_2 * Float64(1.0 + Float64(0.5 * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = x - (y * 2.0);
	t_2 = t_1 / hypot(x, (y * 2.0));
	tmp = 0.0;
	if (t_0 <= 2000.0)
		tmp = t_2 * (1.0 + ((y * 2.0) / x));
	elseif (t_0 <= 5e+210)
		tmp = (t_1 * (x + (y * 2.0))) / (t_0 + (x * x));
	else
		tmp = t_2 * (1.0 + (0.5 * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2000.0], N[(t$95$2 * N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+210], N[(N[(t$95$1 * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1.0 + N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e3

   1. Initial program 57.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 57.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 57.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 25.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 50.6%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 50.6%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 50.6%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 25.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 57.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.3%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.3%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 48.0%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.0%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 48.0%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   if 2e3 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e210

   1. Initial program 81.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 81.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 81.6%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 81.6%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 26.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 35.2%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 35.2%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 35.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 26.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 81.6%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.9999999999999998e210 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 15.2%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 15.2%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 7.5%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 7.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 7.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 7.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 7.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 15.2%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 17.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 17.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 17.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 55.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2000:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + \frac{y \cdot 2}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := 0.5 \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 2000:\\ \;\;\;\;1 + \frac{-6}{{\left(\frac{x}{y}\right)}^{2}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + -1\right) \cdot \left(1 + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (* 0.5 (/ x y))))
   (if (<= t_0 2000.0)
     (+ 1.0 (/ -6.0 (pow (/ x y) 2.0)))
     (if (<= t_0 5e+210)
       (/ (* (- x (* y 2.0)) (+ x (* y 2.0))) (+ t_0 (* x x)))
       (* (+ t_1 -1.0) (+ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = 0.5 * (x / y);
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = 1.0 + (-6.0 / pow((x / y), 2.0));
	} else if (t_0 <= 5e+210) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (t_1 + -1.0) * (1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = 0.5d0 * (x / y)
    if (t_0 <= 2000.0d0) then
        tmp = 1.0d0 + ((-6.0d0) / ((x / y) ** 2.0d0))
    else if (t_0 <= 5d+210) then
        tmp = ((x - (y * 2.0d0)) * (x + (y * 2.0d0))) / (t_0 + (x * x))
    else
        tmp = (t_1 + (-1.0d0)) * (1.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = 0.5 * (x / y);
	double tmp;
	if (t_0 <= 2000.0) {
		tmp = 1.0 + (-6.0 / Math.pow((x / y), 2.0));
	} else if (t_0 <= 5e+210) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (t_1 + -1.0) * (1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = 0.5 * (x / y)
	tmp = 0
	if t_0 <= 2000.0:
		tmp = 1.0 + (-6.0 / math.pow((x / y), 2.0))
	elif t_0 <= 5e+210:
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = (t_1 + -1.0) * (1.0 + t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(0.5 * Float64(x / y))
	tmp = 0.0
	if (t_0 <= 2000.0)
		tmp = Float64(1.0 + Float64(-6.0 / (Float64(x / y) ^ 2.0)));
	elseif (t_0 <= 5e+210)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(t_1 + -1.0) * Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = 0.5 * (x / y);
	tmp = 0.0;
	if (t_0 <= 2000.0)
		tmp = 1.0 + (-6.0 / ((x / y) ^ 2.0));
	elseif (t_0 <= 5e+210)
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	else
		tmp = (t_1 + -1.0) * (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2000.0], N[(1.0 + N[(-6.0 / N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+210], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + -1.0), $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e3

   1. Initial program 57.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 57.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 57.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 57.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 25.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 50.6%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 50.6%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 50.6%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 50.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 25.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 57.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 57.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.3%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.3%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 26.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 26.2%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 48.0%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.0%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 48.0%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   10. Taylor expanded in y around 0 77.0%

       \[\leadsto \color{blue}{1 + -6 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 77.0%

          \[\leadsto 1 + \color{blue}{\frac{-6 \cdot {y}^{2}}{{x}^{2}}} \]

       2. associate-/l* 77.0%

          \[\leadsto 1 + \color{blue}{\frac{-6}{\frac{{x}^{2}}{{y}^{2}}}} \]

       3. unpow2 77.0%

          \[\leadsto 1 + \frac{-6}{\frac{\color{blue}{x \cdot x}}{{y}^{2}}} \]

       4. unpow2 77.0%

          \[\leadsto 1 + \frac{-6}{\frac{x \cdot x}{\color{blue}{y \cdot y}}} \]

       5. times-frac 83.0%

          \[\leadsto 1 + \frac{-6}{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} \]

       6. unpow2 83.0%

          \[\leadsto 1 + \frac{-6}{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}} \]

   12. Simplified 83.0%

       \[\leadsto \color{blue}{1 + \frac{-6}{{\left(\frac{x}{y}\right)}^{2}}} \]

   if 2e3 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e210

   1. Initial program 81.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 81.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 81.6%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 81.6%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 81.6%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 26.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 35.2%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 35.2%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 35.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 35.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 26.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 81.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 81.6%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.9999999999999998e210 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 15.2%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 15.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 15.2%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 7.5%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 7.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 7.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 7.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 7.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 7.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 15.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 15.2%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 15.2%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 17.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 17.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 17.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 17.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 9.3%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 17.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 55.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   8. Taylor expanded in x around 0 88.8%

      \[\leadsto \left(1 + 0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2000:\\ \;\;\;\;1 + \frac{-6}{{\left(\frac{x}{y}\right)}^{2}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y} + -1\right) \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq 19:\\ \;\;\;\;\left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{y \cdot -2}{x}\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1\right) \cdot \left(1 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x y))))
   (if (<= y 19.0)
     (* (+ 1.0 (/ (* y 2.0) x)) (+ 1.0 (/ (* y -2.0) x)))
     (if (<= y 5.7e+109)
       (/ (* (- x (* y 2.0)) (+ x (* y 2.0))) (+ (* y (* y 4.0)) (* x x)))
       (* (+ t_0 -1.0) (+ 1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x / y);
	double tmp;
	if (y <= 19.0) {
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	} else if (y <= 5.7e+109) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x));
	} else {
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x / y)
    if (y <= 19.0d0) then
        tmp = (1.0d0 + ((y * 2.0d0) / x)) * (1.0d0 + ((y * (-2.0d0)) / x))
    else if (y <= 5.7d+109) then
        tmp = ((x - (y * 2.0d0)) * (x + (y * 2.0d0))) / ((y * (y * 4.0d0)) + (x * x))
    else
        tmp = (t_0 + (-1.0d0)) * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x / y);
	double tmp;
	if (y <= 19.0) {
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	} else if (y <= 5.7e+109) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x));
	} else {
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x / y)
	tmp = 0
	if y <= 19.0:
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x))
	elif y <= 5.7e+109:
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x))
	else:
		tmp = (t_0 + -1.0) * (1.0 + t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x / y))
	tmp = 0.0
	if (y <= 19.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * 2.0) / x)) * Float64(1.0 + Float64(Float64(y * -2.0) / x)));
	elseif (y <= 5.7e+109)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) * Float64(x + Float64(y * 2.0))) / Float64(Float64(y * Float64(y * 4.0)) + Float64(x * x)));
	else
		tmp = Float64(Float64(t_0 + -1.0) * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x / y);
	tmp = 0.0;
	if (y <= 19.0)
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	elseif (y <= 5.7e+109)
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x));
	else
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 19.0], N[(N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e+109], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + -1.0), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 19

   1. Initial program 56.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 56.7%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 56.7%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 56.7%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 56.7%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 56.7%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 17.6%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 36.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 36.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 36.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 36.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 36.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 17.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 56.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 56.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 56.7%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.0%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.0%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 17.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 17.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 35.9%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 35.9%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 35.9%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 35.9%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   10. Taylor expanded in x around inf 62.7%

       \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \color{blue}{\left(1 + -2 \cdot \frac{y}{x}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 62.7%

          \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \color{blue}{\frac{-2 \cdot y}{x}}\right) \]

       2. *-commutative 62.7%

          \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{\color{blue}{y \cdot -2}}{x}\right) \]

   12. Simplified 62.7%

       \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \color{blue}{\left(1 + \frac{y \cdot -2}{x}\right)} \]

   if 19 < y < 5.7000000000000002e109

   1. Initial program 68.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 68.4%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 68.4%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 68.4%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 68.4%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 68.4%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 67.8%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 68.4%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 68.4%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 68.4%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 68.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 68.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 67.8%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 68.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 68.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 68.4%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 5.7000000000000002e109 < y

   1. Initial program 12.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 12.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 12.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 12.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 12.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 12.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 11.9%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 12.0%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 12.0%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 12.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 11.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 12.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 12.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 14.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 14.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 14.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 14.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 14.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 14.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 14.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 14.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 14.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 14.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 14.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 14.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 86.9%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   8. Taylor expanded in x around 0 86.7%

      \[\leadsto \left(1 + 0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 19:\\ \;\;\;\;\left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{y \cdot -2}{x}\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y} + -1\right) \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := 0.5 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq 115:\\ \;\;\;\;\left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{y \cdot -2}{x}\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + -1\right) \cdot \left(1 + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (* 0.5 (/ x y))))
   (if (<= y 115.0)
     (* (+ 1.0 (/ (* y 2.0) x)) (+ 1.0 (/ (* y -2.0) x)))
     (if (<= y 5.7e+109)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (* (+ t_1 -1.0) (+ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = 0.5 * (x / y);
	double tmp;
	if (y <= 115.0) {
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	} else if (y <= 5.7e+109) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (t_1 + -1.0) * (1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = 0.5d0 * (x / y)
    if (y <= 115.0d0) then
        tmp = (1.0d0 + ((y * 2.0d0) / x)) * (1.0d0 + ((y * (-2.0d0)) / x))
    else if (y <= 5.7d+109) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (t_1 + (-1.0d0)) * (1.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = 0.5 * (x / y);
	double tmp;
	if (y <= 115.0) {
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	} else if (y <= 5.7e+109) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (t_1 + -1.0) * (1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = 0.5 * (x / y)
	tmp = 0
	if y <= 115.0:
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x))
	elif y <= 5.7e+109:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = (t_1 + -1.0) * (1.0 + t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(0.5 * Float64(x / y))
	tmp = 0.0
	if (y <= 115.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * 2.0) / x)) * Float64(1.0 + Float64(Float64(y * -2.0) / x)));
	elseif (y <= 5.7e+109)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(t_1 + -1.0) * Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = 0.5 * (x / y);
	tmp = 0.0;
	if (y <= 115.0)
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	elseif (y <= 5.7e+109)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = (t_1 + -1.0) * (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 115.0], N[(N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e+109], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + -1.0), $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 115

   1. Initial program 56.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 56.7%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 56.7%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 56.7%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 56.7%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 56.7%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 17.6%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 36.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 36.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 36.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 36.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 36.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 17.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 56.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 56.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 56.7%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.0%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.0%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 17.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 17.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 35.9%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 35.9%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 35.9%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 35.9%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   10. Taylor expanded in x around inf 62.7%

       \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \color{blue}{\left(1 + -2 \cdot \frac{y}{x}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 62.7%

          \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \color{blue}{\frac{-2 \cdot y}{x}}\right) \]

       2. *-commutative 62.7%

          \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{\color{blue}{y \cdot -2}}{x}\right) \]

   12. Simplified 62.7%

       \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \color{blue}{\left(1 + \frac{y \cdot -2}{x}\right)} \]

   if 115 < y < 5.7000000000000002e109

   1. Initial program 68.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   if 5.7000000000000002e109 < y

   1. Initial program 12.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 12.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 12.0%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 12.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 12.0%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 12.0%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 11.9%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 12.0%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 12.0%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 12.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 11.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 12.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 12.0%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 12.0%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 14.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 14.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 14.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 14.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 14.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 14.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 14.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 14.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 14.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 14.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 14.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 14.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 86.9%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   8. Taylor expanded in x around 0 86.7%

      \[\leadsto \left(1 + 0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 115:\\ \;\;\;\;\left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{y \cdot -2}{x}\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y} + -1\right) \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq 3600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1\right) \cdot \left(1 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x y))))
   (if (<= y 3600.0) 1.0 (* (+ t_0 -1.0) (+ 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x / y);
	double tmp;
	if (y <= 3600.0) {
		tmp = 1.0;
	} else {
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x / y)
    if (y <= 3600.0d0) then
        tmp = 1.0d0
    else
        tmp = (t_0 + (-1.0d0)) * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x / y);
	double tmp;
	if (y <= 3600.0) {
		tmp = 1.0;
	} else {
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x / y)
	tmp = 0
	if y <= 3600.0:
		tmp = 1.0
	else:
		tmp = (t_0 + -1.0) * (1.0 + t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x / y))
	tmp = 0.0
	if (y <= 3600.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(t_0 + -1.0) * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x / y);
	tmp = 0.0;
	if (y <= 3600.0)
		tmp = 1.0;
	else
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3600.0], 1.0, N[(N[(t$95$0 + -1.0), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3600

   1. Initial program 56.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 61.5%

      \[\leadsto \color{blue}{1} \]

   if 3600 < y

   1. Initial program 27.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 27.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 27.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 27.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 27.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 27.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 27.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 27.5%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 27.5%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 27.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 27.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 27.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 29.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 29.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 29.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 29.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 29.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 29.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 29.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 29.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 29.5%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 29.5%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 29.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 29.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 80.9%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   8. Taylor expanded in x around 0 80.6%

      \[\leadsto \left(1 + 0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y} + -1\right) \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq 27000:\\ \;\;\;\;\left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{y \cdot -2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1\right) \cdot \left(1 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x y))))
   (if (<= y 27000.0)
     (* (+ 1.0 (/ (* y 2.0) x)) (+ 1.0 (/ (* y -2.0) x)))
     (* (+ t_0 -1.0) (+ 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x / y);
	double tmp;
	if (y <= 27000.0) {
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	} else {
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x / y)
    if (y <= 27000.0d0) then
        tmp = (1.0d0 + ((y * 2.0d0) / x)) * (1.0d0 + ((y * (-2.0d0)) / x))
    else
        tmp = (t_0 + (-1.0d0)) * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x / y);
	double tmp;
	if (y <= 27000.0) {
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	} else {
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x / y)
	tmp = 0
	if y <= 27000.0:
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x))
	else:
		tmp = (t_0 + -1.0) * (1.0 + t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x / y))
	tmp = 0.0
	if (y <= 27000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * 2.0) / x)) * Float64(1.0 + Float64(Float64(y * -2.0) / x)));
	else
		tmp = Float64(Float64(t_0 + -1.0) * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x / y);
	tmp = 0.0;
	if (y <= 27000.0)
		tmp = (1.0 + ((y * 2.0) / x)) * (1.0 + ((y * -2.0) / x));
	else
		tmp = (t_0 + -1.0) * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 27000.0], N[(N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + -1.0), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 27000

   1. Initial program 56.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 56.7%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 56.7%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 56.7%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 56.7%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 56.7%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 17.6%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 36.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 36.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 36.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 36.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 36.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 17.6%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 56.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 56.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 56.7%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 58.0%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 58.0%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 58.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 58.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 58.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 17.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 17.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 17.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 58.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 58.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around 0 35.9%

      \[\leadsto \color{blue}{\left(1 + 2 \cdot \frac{y}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 35.9%

         \[\leadsto \left(1 + \color{blue}{\frac{2 \cdot y}{x}}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

      2. *-commutative 35.9%

         \[\leadsto \left(1 + \frac{\color{blue}{y \cdot 2}}{x}\right) \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   9. Simplified 35.9%

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot 2}{x}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   10. Taylor expanded in x around inf 62.7%

       \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \color{blue}{\left(1 + -2 \cdot \frac{y}{x}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 62.7%

          \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \color{blue}{\frac{-2 \cdot y}{x}}\right) \]

       2. *-commutative 62.7%

          \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{\color{blue}{y \cdot -2}}{x}\right) \]

   12. Simplified 62.7%

       \[\leadsto \left(1 + \frac{y \cdot 2}{x}\right) \cdot \color{blue}{\left(1 + \frac{y \cdot -2}{x}\right)} \]

   if 27000 < y

   1. Initial program 27.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 27.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 27.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 27.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 27.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 27.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 27.3%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 27.5%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 27.5%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 27.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 27.3%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 27.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 27.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 29.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 29.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-def 29.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 29.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-def 29.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. sqrt-prod 29.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. *-commutative 29.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot y}} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 29.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{4} \cdot \sqrt{y}\right)} \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 29.5%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left(\color{blue}{2} \cdot \sqrt{y}\right) \cdot \sqrt{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*r* 29.5%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 29.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{y}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      13. *-commutative 29.8%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 80.9%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   8. Taylor expanded in x around 0 80.6%

      \[\leadsto \left(1 + 0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 27000:\\ \;\;\;\;\left(1 + \frac{y \cdot 2}{x}\right) \cdot \left(1 + \frac{y \cdot -2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y} + -1\right) \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 21000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 21000.0) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 21000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 21000.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 21000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 21000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 21000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 21000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 21000.0], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 21000

   1. Initial program 56.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 61.5%

      \[\leadsto \color{blue}{1} \]

   if 21000 < y

   1. Initial program 27.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 80.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 21000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 48.8%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 49.5%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 49.5%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 50.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024018 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))